You want to bake cupcakes for your instructor! You need a recipe and get the ing

Question

You want to bake cupcakes for your instructor! You need a recipe and get the ingredients ready. The butter and eggs came from the refrigerator, and the dry ingredients are at room temperature, which is 75F. When you mix everything together, the temperature of the cupcake batter is 60F. You were in such a hurry to make cupcakes that you forgot to pre-heat the oven. But you don’t want to wait any longer, so you turn it on when you put the 60F cupcakes in the oven. The temperature is initially 75F in the oven, just like it is in your kitchen, and then it increases by 25F per minute until it reaches 350F. Then it just stays at 350F. Assume that the rate of change of the temperature u(t) of the cupcakes in the oven is equal to 1 times the difference between u(t) and the oven temperature.

Using the Laplace transform, find the temperature u(t) of the cupcakes in the oven at any time t.

T(t) = 75 + 25t if 0 t < 11

350 if t 11.

Sol. First note that it takes 11 minutes for the temperature of the oven to go from 75°F to 350°F. It follows that the temperature T() of the oven at time t is given by 75 25t if 0 t 11 50 if t 11. and u satisfies the initial value problem u' u First we compute the Laplace transform of T. Note that we have +350uu (t) 75 +25t 25(t 11) u11 (t), so that 25e 118 (s) 75- 25 Now, taking the Laplace transform of the equation, we get 118 F(s) 60 F(s) 755 252 25e

Explanation / Answer

u'+u = T
This equation hold true because it is given itself in the question in the satement "Assume that the rate of change of the temperature u(t) of the cupcakes in the oven is equal to 1 times the difference between u(t) and the oven temperature." which can be written as

u'(t) = -[u(t) - T]

which can be rearranged and written as u'(t) + u(t) = T.

Where u'(t) is the rate of change in tempreture of cupcakes, u(t) is the tempreture of cupcake at time t, and T is the tempreture of oven.

It is clearly mentioned in that statement to assume it.

This equation is then used as initial value problem where both the dericative u'(t) and initial value of u at time t=0 is know (i.e 60).

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